Which of the following is equivalent to sec β sin ( α + β ) for all values of α and β for which sec β sin ( α + β ) is defined?

Question

Which of the following is equivalent to secβsin(α+β) for all values of α and β for which secβsin(α+β) is defined?

Answer 1

secbsin(a+b)

=secb(sinacosb + sinbcosa)

=secbsinacosb + secbsinbcosa

(1/cosb)sinacosb + (1/cosb)sinbcosa

=sina + (sinb/cosb)cosa

= sina+tanbcosa

Answer 2

First, recall that secβ=1/cosβ so we can write:

secβsin(α+β)=sin(α+β)​/cosβ

Next, use the angle addition formula for sine:

sin(α+β)=sinαcosβ+cosαsinβ

Substituting this into the expression, we get:

sin(α+β)​/cosβ=(sinαcosβ+cosαsinβ)​/cosβ

Now, split up the fraction:

(sinαcosβ+cosαsinβ)​/cosβ=(sinαcosβ)​/cosβ+(cosαsinβ)​/cosβ

This simplifies to:

sinα+cosαtanβ

Hope this was helpful.

Answer 3

Where are the choices you referenced?